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Acta Physica Sinica

A THREE-PHOTON VECTOR MODEL

Peixian Ye, 中国科学院物理研究所

Published Jan 1, 1979• 0 citations
Publisher

Abstract

It is shown that the optical Eabi precession model developed by Feynman, Vernon and Hellwarth for two-level systems can be extended not only to discuss the two-photon resonant coherent interaction, but also to the case of three photons. Using this model and the adiabatic following approximation, the third-order polarization expression in an intense and (or) resonant optical field is obtained, which differs from that obtained by perturbation theory. The saturation phenomena in the third-order resonant non-linear effects are also discussed.

DOI
10.7498/aps.28.32

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It is shown that the optical Eabi precession model developed by Feynman, Vernon and Hellwarth for two-level systems can be extended not only to discuss the two-photon resonant coherent interaction, but also to the case of three photons. Using this model and the adiabatic following approximation, the third-order polarization expression in an intense and (or) resonant optical field is obtained, which differs from that obtained by perturbation theory. The saturation phenomena in the third-order resonant non-linear effects are also discussed.
Two-photon resonantly enhanced parametric generation processes have generally been described using time-dependent perturbation theory. In this paper we show that a theory of two-photon coherent effects can be used to derive and explain these nonlinear mixing processes. Our technique makes use of the adiabatic following (AF) approximation to obtain solutions to a vector model describing the two-photon resonance. We show that the usual results for the nonlinear susceptibilities correspond to the $\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}$ vector of Feynman, Vernon, and Hellwarth adiabatically following the $\stackrel{\ensuremath{\rightarrow}}{\mathrm{\ensuremath{\gamma}}}$ vector in the small-angle limit. Consequently, the theory allows a natural extension to large angles, and power-dependent nonlinear susceptibilities are obtained. We then use these AF results for the polarization to study the propagation of pulses nearly resonant with a two-photon transition, and we demonstrate that the pulse reshaping is due to the two related effects of a nonlinear pulse velocity and self-phase modulation.
We propose and develop a method for theoretical description of circularly (elliptically) polarized optical pulse resonant coherent interactions with two-level atoms. The method is based on the time-evolution equations of a two-level quantum system in the presence of a time-dependent dipole perturbation for electric dipole transitions between states with total angular-momentum projection difference $(\ensuremath{\Delta}{J}_{z}=\ifmmode\pm\else\textpm\fi{}1)$ excited by a circularly polarized electromagnetic field [Feynman et al., J. Appl. Phys. 28, 49 (1957)]. The adopted real-vector representation approach allows for coupling with the vectorial Maxwell's equations for the optical wave propagation and thus the resulting Maxwell pseudospin equations can be numerically solved in the time domain without any approximations. The model permits a more exact study of the ultrafast coherent pulse propagation effects taking into account the vector nature of the electromagnetic field and hence the polarization state of the optical excitation. We demonstrate self-induced transparency effects and formation of polarized solitons. The model represents a qualitative extension of the well-known optical Maxwell-Bloch equations valid for linearly polarized light and a tool for studying coherent quantum control mechanisms.
We derive an exact, intensity-dependent expression for the susceptibility of a three-level system interacting with one monochromatic electric field. This expression, once expanded, is equivalent to the usual perturbative series χ=χ(1)+χ(3)E2+⋅⋅⋅ within the rotating wave approximation. We consider two types of resonant processes: one- and two-photon resonances corresponding to the intensity-dependent susceptibility χ(3)(−ω;ω,−ω,ω), and one-, two-, and three-photon resonances corresponding to χ(3)(−3ω;ω,ω,ω). As an example of current interest, we use a model three-level system that mimics the excited electronic states of typical nonlinear optical polymers and show that near resonance, successive terms in the perturbative series approach the same order of magnitude for experimentally realizable fields.
A theory of resonantly enhanced degenerate four-wave mixing in two-level systems including the effects of atomic and photon angular momentum is presented in the absence of pump-induced saturation of the transition. It is shown that there exist three distinct quantum-mechanical amplitudes leading to the third-order polarization density. These quantum-mechanical amplitudes are shown to be sensitive to the states of polarization of the incident fields. The quantum-mechanical transport equation in the mmm representation is used to calculate the output signal in the various regimes of laser detuning, atomic linewidth, and polarization states of the radiation field for collinear interaction.
The non-linear optical polarization of an isolated atom or molecule is treated, giving careful consideration to secular and resonant terms in the perturbation expansion. The Method of Averages introduced by Bogoliubov and Mitropolsky is used. The case where resonance-induced excited state populations are negligible, which is relevant to a wide range of non-linear optical experiments, is examined in detail for polarizations through third order in the perturbing fields. This yields concise expressions which are valid for any combination of applied field frequencies, including static fields.
The optical-pumping cycle in the three-level system e,g,f (J e =1=J f , J g =0), irradiated by two monochromatic sigma -polarized laser light waves is analysed theoretically by the successive approximation (iteration) method up to fourth order. The solutions of the density-matrix evolution equations describe (i) in second order, the influence of the Zeeman structure of the e and f states on the saturation resonances in the intermediate level g (second-order population effect) and (ii) in fourth order, the influence of the Hanle effects and the saturation phenomena on the shape of the double-quantum Hanle resonances and on the two-photon transition probabilities.